How to extract commands such as single dupliverts
The row of "dupli_type" options [None, Frame, Verts, Faces, Group] comes from what is called an EnumProperty. Blender uses this as a radio-box-like interface to give you a choice from a list. The item you select determines which elements of the UI become visible.


  Right-Click on the Verts button, pick Edit Source, go to the TextEditor and open 
  properties_object.py it will show at the line where that UI element is  referenced. 


The Duplication panel's draw is like this. Notice that the 5 possible states are covered by the if-elif-elif-elif. Notably the choice for NONE results in no drawing at all because it isn't in any of those tests. There's nothing to show in the case of 'NONE' so a 5th logic statement is redundant.

def draw(self, context):
    layout = self.layout

    ob = context.object

    layout.prop(ob, "dupli_type", expand=True)

    if ob.dupli_type == 'FRAMES':
        split = layout.split()

        col = split.column(align=True)
        col.prop(ob, "dupli_frames_start", text="Start")
        col.prop(ob, "dupli_frames_end", text="End")

        col = split.column(align=True)
        col.prop(ob, "dupli_frames_on", text="On")
        col.prop(ob, "dupli_frames_off", text="Off")

        layout.prop(ob, "use_dupli_frames_speed", text="Speed")

    elif ob.dupli_type == 'VERTS':
        layout.prop(ob, "use_dupli_vertices_rotation", text="Rotation")

    elif ob.dupli_type == 'FACES':
        row = layout.row()
        row.prop(ob, "use_dupli_faces_scale", text="Scale")
        sub = row.row()
        sub.active = ob.use_dupli_faces_scale
        sub.prop(ob, "dupli_faces_scale", text="Inherit Scale")

    elif ob.dupli_type == 'GROUP':
        layout.prop(ob, "dupli_group", text="Group")


To make a custom panel for this you need to make a custom (deputy) EnumProperty, which should include some code to update the real property.

import bpy


class HelloWorldPanel(bpy.types.Panel):
    """Creates a Panel in the Object properties window"""
    bl_label = "Hello World Panel"
    bl_idname = "OBJECT_PT_hello"
    bl_space_type = 'PROPERTIES'
    bl_region_type = 'WINDOW'
    bl_context = "object"

    def draw(self, context):
        layout = self.layout
        ob = context.object

        layout.prop(ob, "deputy_dupli_type", expand=True)

        if ob.deputy_dupli_type == 'VERTS':
            layout.prop(ob, "use_dupli_vertices_rotation", text="Rotation")

        elif ob.deputy_dupli_type == 'FACES':
            row = layout.row()
            row.prop(ob, "use_dupli_faces_scale", text="Scale")
            sub = row.row()
            sub.active = ob.use_dupli_faces_scale
            sub.prop(ob, "dupli_faces_scale", text="Inherit Scale")


def register():
    # give all objects a new property (also known as an attribute)
    # called "deputy_duply_type".
    bpy.types.Object.deputy_dupli_type = bpy.props.EnumProperty(
        items=[
            ('NONE', 'NONE', "", 0),
            ('VERTS', 'VERTS', "", 1),
            ('FACES', 'FACES', "", 2)                        
        ],
        default='NONE',
        update=lambda s, c: setattr(s, 'dupli_type', s.deputy_dupli_type)
    )
    bpy.utils.register_class(HelloWorldPanel)


def unregister():
    bpy.utils.unregister_class(HelloWorldPanel)
    del bpy.types.Object.deputy_dupli_type


if __name__ == "__main__":
    register()


The downside in this case is that changes made back in the original Duplication Panel are not reflected in the state of your new deputy_dupli_type property. This may lead to some confusion, but if you aren't using the original panels then it isn't a problem.

Some code clarification:  

update=lambda s, c: setattr(s, 'dupli_type', s.deputy_dupli_type)


The take-away from this line is (even if it looks like gibberish): 


Once the user picks a different option from the object.deputy_dupli_type, this triggers the object's dupli_type attribute to be set to the same value as the newly selected option.